# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region y=2-x, 2<=x<=4 rotated about the x-axis?

Oct 9, 2015

Always graph the constraints first, then determine the correct integration formula.

#### Explanation:

Here is a graph of the constraints:

Now, without using calculus, you should visualize rotating this line around the x-axis will generate a cone with radius 2 and height 2. The volume of a cone is:

$V = \left(\frac{1}{3}\right) \pi {r}^{2} h = \left(\frac{1}{3}\right) \pi {2}^{2} \cdot 2 = \left(\frac{8}{3}\right) \pi$

So, in advance of using calculus, we know the answer is $\left(\frac{8}{3}\right) \pi$

Now, using calculus and the shell method :

Because we rotate around the x-axis, integrate with respect to y:

$2 \pi {\int}_{-} {2}^{0} y \left[4 - \left(2 - y\right)\right] \mathrm{dy} = \left(\frac{8}{3}\right) \pi$

hope that helped